271 years after the first Beis Hamikdash's foundations were laid, a new era in its history began, with Chizkiyahu's purifying it from the idolatry introduced by his father Achaz and urging the Jews to make it the center of their lives (II Chron. chs. 29-31).

(Construction on the Beis Hamikdash began in the fourth year of Shlomo's reign, in 2928 since Creation. Chizkiyahu began his program of improving Jewish life immediately on his accession to the throne, in 3199.)

271 is the numerical value of "Ani HaGever" (I am the man) and Yirmiyahu, who is the author of Eichah, and who laments, in 3:1, that he had to not only prophesy, but also "witnessed affliction by the rod of (HaShem's) anger."
(Artscroll Eichah pg. 33, note 1)